• Note on a problem of Ramanujan

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/131/0019

• # Keywords

Lattice points; irrationality measure; diophantine approximation.

• # Abstract

For fixed positive real numbers $\omega , \omega '$, it is known that the number of lattice points $(u,v), u\ge 0, v \ge 0$ satisfying $0 \le u \omega + v\omega ' \le \eta$ is given by $\frac{1}{2}\big (\frac{\eta ^{2}}{\omega \omega ^{'}}+\frac{\eta }{\omega } +\frac{\eta }{\omega ^{'}}\big )+ O_{\varepsilon }(\eta ^{1-\frac{1}{\alpha _{0}} +\varepsilon })$, where $\alpha _0 \ge 1$ is a constant. In this paper, we explicitly compute $\alpha _0$ for certain values of $\omega /\omega '$. In particular, in Ramanujan's case (i.e., when $\omega = \log 2$ and $\omega ' = \log 3$), we show that $\alpha _0 = 2^{18}\log 3$ is admissible. This improves an earlier result of the paper (Ramachandra K, Sankaranarayanan A and Srinivas K, Hardy Ramanujan J. 19 (1996) 2--56), where it was shown that $\alpha _0 = 2^{40}\log 3$ holds.

• # Author Affiliations

1. Institute of Mathematical Sciences, HBNI, C.I.T. Campus, Taramani, Chennai 600 113, India
2. SRTM University, Vishnupuri, Nanded 431 606, India

• # Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019